1. Field of the Invention
This invention relates generally to reconstruction of the density function of a three-dimensional object from a set of cone-beam projections, such as from an X-ray source. More particularly, the invention relates to methods for cone beam reconstruction using backprojection of locally filtered projections and an X-ray computed tomography (CT) apparatus incorporating the method.
2. Description of Related Art
Technology for X-ray detection in cone-beam (CB) geometry is rapidly improving and offers more and more potential for the construction of robust computed tomography (CT) systems for fast high-resolution volume imaging. However, to optimally build such systems, the problem of CB image reconstruction needs to be fully understood.
The successive works of H. Tuy, “An Inversion Formula for Cone-Beam Reconstruction,” SIAM J. Appl. Math., No. 43, pp. 546-52, 1983, B. D. Smith, “Image Reconstruction from Cone-Beam Projections: Necessary and Sufficient Conditions and Reconstruction Methods,” IEEE Trans. Med. Imag., Vol. MI-4, pp. 1425, 1985 and P. Grangeat, “Mathematical Framework of Cone-Beam 3D Reconstruction via the First Derivative of the Radon Transform,” in Mathematical Methods in Tomography, G. T. Herman, A. K, Louis, and F. Natterer, Eds. Berlin, Germany: Springer-Verlag, 1991, Vol. 1497, Lecture Notes in Mathematics, pp. 66-97, have shown that exact reconstruction at a given location is possible if every plane passing through that location intersects the trajectory of the X-ray source. However, this well-known result, which is fundamental and clearly represented a breakthrough, is not sufficient for most practical applications because its derivation assumed complete CB projections. When only part of the imaged object is illuminated at a given source position, the CB projection is said to be incomplete or truncated. Exact and accurate reconstruction from truncated projections is more complicated than from complete projections. An overview of the problem of reconstructing from truncated projections may be found in R. Clackdoyle, M. Defrise and F. Noo, “Early Results on General Vertex Sets and Truncated Projections in Cone-Beam Tomography,” in Computational Radiology and Imaging: Therapy and Diagnostics, C. Brgers and F. Natterer, Eds. Berlin, Germany: Springer-Verlag, 1999, Vol. 110, IMA Volumes in Mathematics and Its Applications, pp. 113-135. Under some conditions, the problem of reconstructing from truncated projections may even be impossible, see e.g., F. Natterer, The Mathematics of CT, Philadelphia, Pa.: SIAM, 2001.
A general theory to handle CB data truncation remains elusive. Solutions have been found only for particular measurement geometries. Many of these solutions were obtained using a clever handling of data redundancy in a CB filtered-backprojection (FBP) reconstruction framework. See e.g., H. Kudo and T. Saito, “An Extended Completeness Condition for Exact Cone-Beam Reconstruction and Its Applications,” in Conf. Rec. 1994 IEEE Nuclear Science Symp. Medical Imaging Conf., Norfolk Va., 1995, pp. 1710-14; F. Noo, R. Clackdoyle, and M. Defrise, “Direct Reconstruction of Cone-Beam Data Acquired with a Vertex Path Containing a Circle,” IEEE Trans, Image Process., Vol. 7, No. 6, pp. 854-67, Jun. 1998; R. H. Johnson, H. Hu, S. T. Haworth, P. S. Cho, C. A. Dawson and J. H. Linehan, “Feldkamp and Circle and Line Cone-Beam Reconstruction for 3D Micro-CT of Vascular Networks,” Phys. Med. Biol. Vol. 43, pp. 929-40, 1998; H. Kudo and T. Saito, “Fast and Stable Cone-Beam Filtered Backprojection Method for Nonplanar Orbits,” Phys. Med. Biol., Vol. 43, pp. 747-60, 1998; H. Kudo, F. Noo and M. Defrise, “Quasi-Exact Filtered Backprojection Algorithm for Long-Object Problem in Helical Cone-Beam Tomography,” IEEE Trans, Med. Imag., Vil. 19, No. 9, pp. 902-21, September 2000; G. Lauritsch, K. C. Tam, K. Sourbelle and S. Schaller, “Exact Local Region-of-Interest Reconstruction in Spiral Cone-Beam Filtered-Backprojection CT: Numerical Implementation and First Image Results,” in Proc. SPIE Medical Imaging Conf. (Image Processing), Vol. 3979, 2000, pp. 520-32; K. C. Tam, G. Lauritsch and K. Sourbelle, “Filtering Point Spread Function in Backprojection Cone-Beam CT and Its Applications in Long Object Imaging,” Phys, Med. Biol., Vol. 47, pp. 2685-703, 2002; A. Katesevich, “A General Scheme for Constructing Inversion Algorithms for Cone-Beam CT,” I.J.M.M.S., Vol. 21, pp. 1305-21, 2003; G. H. Chen, “An Alternative Derivation of Katsevich's Cone-Beam Reconstruction Formula,” Med. Phys., Vol. 30, No. 12, pp. 3217-26, 2003; J. D. Pack, F. Noo and H. Kudo, “Investigation of Saddle Trajectories for Cardiac CT Imaging in Cone-Beam Geometry,” Phys. Med. Biol., Vol. 49, No. 11, pp. 2317-36; and H. Kudo, F. Noo and M. Defrise, “Cone-Beam Filtered-Backprojection Algorithm for Truncated Helical Data,” Phys. Med. Biol., Vol. 43, pp. 2885-909, 1998.
Other solutions for CB reconstruction from truncated projections for particular measurement geometries were obtained using the formula posed by Grangeat or its truncated version, see H. Kudo, F. Noo, and M. Defrise, “Cone-Beam Filtered-Backprojection Algorithm for Truncated Helical Data,” Phys. Med. Biol., Vol. 43, pp. 2885-909, 1998, in combination with properties of the three-dimensional (3-D) radon transform. See, e.g., H. Kudo and T. Saito, “Extended Cone-Beam Reconstruction Using Radon Transform,” in Conf Rec. 1996 IEEE Nuclear Science Symp. Medical Imaging Conf., Anaheim, Calif., 1997, pp. 1693-97; K. C. Tam, S. Samarasekera and F. Sauer, “Exact Cone-Beam CT with a Spiral Scan,” Phys. Med. Biol, Vol. 43, pp. 1015-24, 1998; S. Schaller, F. Noo, F. Sauer, K. C. Tam, G. Lauritsch and T. Flohr, “Exact Radon Rebinning Algorithm for the Long Object Problem in Helical Cone-Beam CT,” IEEE Trans, Med. Imag., Vol. 19, No. 5, pp. 36 1-75, May 2000; and X. Tang and R. Ning, “A Cone Beam Filtered Backprojection (CB-FBP) Reconstruction Algorithm for a Circle-Plus-Two-Arc Orbit,” Med. Phys., Vol. 28, No. 6, pp. 1042-55, 2001.
Many advances in CB reconstruction have been made recently in reconstruction methods for use in helical CB tomography (HCBT). Two of the most interesting results achieved in the HCBT context are disclosed in A. I. Katsevich, “An Improved Exact Filtered Backprojection Algorithm for Spiral Computed Tomography,” Adv. Appl. Math., Vol. 32, No. 4, pp. 681-97, 2004 and Y. Zou and X. Pan, “Exact Image Reconstruction on PI Lines from Minimum Data in Helical Cone-Beam CT,” Phys. Med. Biol., Vol. 49, pp. 941-59, 2004.
Katsevich showed that exact and accurate FBP reconstruction can be achieved with a simple one-dimensional (1-D) Hilbert transform of a derivative of the projection, using a minimum overscan and just slightly more than the data in the Tam-Danielsson (TD) window, see e.g., K. C. Tam, S. Samarasekera and F. Sauer, “Exact Cone-Beam CT with a Spiral Scan,” Phys. Med. Biol, Vol. 43, pp. 1015-24, 1998; and P. E. Danielsson, P. Edholm, J. Eriksson and M. Magnusson Seger, “Toward Exact Reconstruction for Helical Cone-Beam Scanning of Long Objects: A New Detector Arrangement and a New Completeness Condition,” in Proc. 1997 Meeting Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, D. W. Townsend and P. E. Kinahan, Eds., Pittsburgh, Pa., 1997, pp. 141-44. Katsevich has also generalized his formula to general CB tomography and showed incidentally that his formula belongs to the family of methods that can be obtained using a clever handling of data redundancy in a CB-FBP reconstruction framework, see e.g., A. Katesevich, “A General Scheme for Constructing Inversion Algorithms for Cone-Beam CT,” I.J.M.M.S., Vol. 21, pp. 1305-21, 2003.
Zou and Pan investigated the effect of skipping the 1-D Hilbert transform in the reconstruction steps of Katsevich's formula. They argued that by so doing the outcome of the backprojection on any π-line is the Hilbert transform of the values of the density function, ƒ, on this π-line, and they devised from this argument and information on the support of ƒ an accurate algorithm that has the same overscan and efficiency as Katsevich's formula but uses only the data in the TD window. Recall that a π-line is any line segment that connects two points of the helix separated by less than one helix turn, see e.g., P. E. Danielsson, P. Edholm, J. Eriksson and M. Magnusson Seger, “Toward Exact Reconstruction for Helical Cone-Beam Scanning of Long Objects. A New Detector Arrangement and a New Completeness Condition,” in Proc. 1997 Meeting Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, D. W. Townsend and P. E. Kinahan, Eds., Pittsburgh, Pa., 1997, pp. 141-44.
None of these conventional approaches appears to achieve CB reconstruction on various measured lines using virtually arbitrary source trajectories. A measured line is any line that contains a source position and is part of the measurements. Thus, it would be highly advantageous to provide a method for CB reconstruction using backprojection of locally filtered projections and an X-ray CT apparatus incorporating such a method.